By-passing fluctuation theorems with a catalyst
P. Boes
1, R. Gallego
1, N. H. Y. Ng
1, J. Eisert
1, and H. Wilming
21Dahlem Center for Complex Quantum Systems, Freie Universit¨at Berlin, 14195 Berlin, Germany 2Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland
Fluctuation theorems impose constraints on possible work extraction probabilities in ther-modynamical processes. These constraints are stronger than the usual second law, which is
concerned only with average values. Here,
we show that such constraints, expressed in the form of the Jarzysnki equality, can be by-passed if one allows for the use of catalysts— additional degrees of freedom that may be-come correlated with the system from which work is extracted, but whose reduced state re-mains unchanged so that they can be re-used. This violation can be achieved both for small systems but also for macroscopic many-body systems, and leads to positive work extrac-tion per particle with finite probability from macroscopic states in equilibrium. In addition to studying such violations for a single system, we also discuss the scenario in which many par-ties use the same catalyst to induce local tran-sitions. We show that there exist catalytic pro-cesses that lead to highly correlated work dis-tributions, expected to have implications for stochastic and quantum thermodynamics.
1
Introduction
Consider a physical system in thermal equilibrium with its environment. The second law of thermodynamics dic-tates that it is impossible to extract positive average work from this system using reversible processes that are cyclic in the Hamiltonian. More precisely, if the system’s ini-tial state is represented by a canonical ensemble and we consider many iterations of a probabilistic process during which the Hamiltonian of the system is varied but returned to the initial Hamiltonian at the end, then it holds that
hWi ≤0, (1)
wherehWiis the average work extracted during the
pro-cess. We will refer to(1)as theAverage Second Law
(Av-SL),
However, there exist significantly stronger constraints on the possible extracted work in the above type of
pro-cesses, namely those imposed by fluctuation theorems
[1, 2, 3]. Indeed, using such theorems, one can show
that the probability of extracting a finite amount of pos-itive work per particle is exponentially suppressed with
the number of particles in a system [1]. Once these dif-ferent types of constraints are recognized, an interesting questions arises: What are physically meaningful settings in which the probabilistic constraints imposed by fluctua-tion theorems can be circumvented, while still respecting the Av-SL? In particular, do fluctuation theorems also hold when an additional, cyclically evolving auxiliary system is allowed for?
In this work, we present an answer to this question, by introducing a class of processes that generalize the above reversible processes, are physically well motivated,
com-patible with(1), and yet allow for the extraction of positive
work per particle with a probability that is independent of
system size. We do so via the notion of acatalytic process,
in which we allow for the reversible process to not only act on the system as such, but additionally on an auxiliary sys-tem that can be initially prepared in an arbitrary state, but whose marginal state has to be left invariant by the process. Such catalysts are well-motivated – they allow a general description of thermodynamic processes in which the sys-tem may be interacting with some experimental apparatus
(such as a quantum clock [4,5]), however not extracting
energetic/information resources from such an ancilla. In terms of our discussion of the Av-SL above, catalysts cor-respond to the cyclically evolving auxiliary system. De-spite being studied frequently in resource-theoretic
formu-lations of thermodynamics [6,7,8,9], catalytic processes
have never been studied in the context of fluctuation theo-rems until now. Furthermore, even in previous works of catalysis, the exact form of the catalyst is highly
state-dependent and therefore rarely studied explicitly [6, 8].
In this work, we make progress in the significant gaps in the knowledge of catalysis, by presenting and discussing constructive examples of such catalytic processes in the framework where fluctuation theorems are commonly de-rived. We show that, by sharing the same catalyst, a group of agents can follow collective strategies to achieve highly correlated work-distributions. This makes these processes
interesting for the field of quantum and stochastic
ther-modynamics and potentially also for certain negentropic
processes in biology. On the overall, our work provides a rigorous footing for the further study of thermodynamical
processes that systematically exploit the notion of
cataly-sisin order to achieve certain patterns of work fluctuations
in an environment that is governed by the Av-SL. Given the broad applicability of our results, we believe that the study of such processes will produce many further inter-esting results of both foundational and practical interest.
2
Setup
2.1
Formulation of the physical situation
We formulate our arguments and results in the language of quantum mechanics, but all of our results similarly apply to classical, stochastic systems. We consider the setting
depicted in Fig.1: Ad-dimensional systemSwith
Hamil-tonianH = Pdi=1Ei|EiihEi|is initalized in the Gibbs
state
ωβ(H) :=
e−βH Z(β, H),
whereZ(β, H) := Tr(e−βH). This state describes a
sys-tem initially in thermal equilibrium with its environment
at inverse temperature β := 1/(kBT). An agent (some
experimenter) first performs an energy measurement on
this system which produces a measurement outcomeEi.
According to quantum mechanics, the post-measurement
state is described by the density matrix |EiihEi|. The
agent then performs a physical operation on the system which does not depend on the outcome of the measure-ment. Such an operation can always be represented by a
general quantum channelC(i.e., a trace-preserving,
com-pletely positive map that takes density matrices to den-sity matrices) applied to the post-measurement state. This operation is then followed by a second energy measure-ment with respect to the same Hamiltonian with outcome
Ef 1. This procedure results in a channel-dependent joint
distributionP(Ef, Ei) =P(Ef|Ei)P(Ei). In general, a
given quantum channel may be realized in different ways.
Whether the change of energyEf−Eican be interpreted
as work from a thermodynamic point of view will depend
on how exactly the quantum channelCwas physically
re-alized. We will assume that this is the case in the follow-ing, but will comment on this assumption again later on.
In particular, we can then define the work distributionP
for the above process as
P(W) :=X
i,f
P(Ef, Ei)δ(W −(Ei−Ej)),
whereδis the Dirac delta distribution. We are interested in
investigating possible distributionsP(W)that arise from
different channelsC. To do so, it is useful to note the
rela-tion
heβWi=X
j
e−βEj
ZH
hEj| C[I]|Eji, (2)
which is straightforwardly derived using the above
defini-tions, whereIdenotes the identity matrix.
In the standard setting of Tasaki-type fluctuation
the-orems, C is considered to be a unitary channel C[·] =
U(·)U†, since these are generated by changing the
Hamil-tonian over time [3]. For such channels,(2)becomes
heβWi= 1, (3)
1It is possible to extend the setup and our further results
to the more general case of different Hamiltonians for the ini-tial and final measurement. We present our results within this restricted settings for conceptual and notational simplicity.
! (H)
P(Ei)
|EiihEi| C(|EiihEi|)
P<latexit sha1_base64="VClE/+jMejTXnbnkLSNNAFdPOWU=">AAACOnicdVDLSsNAFJ3UV62vVjeCm2AR6qYkKuiyWAouK9gHtCVMppN26EwSZm6EEuvXuNUv8EfcuhO3foCTNgub4oGBwzn3MuceN+RMgWV9GLm19Y3Nrfx2YWd3b/+gWDpsqyCShLZIwAPZdbGinPm0BQw47YaSYuFy2nEn9cTvPFKpWOA/wDSkA4FHPvMYwaAlp3jcrDQc76nhsHMn7gsMY4J5XJ/NnGLZqlpzmKvETkkZpWg6JaPQHwYkEtQHwrFSPdsKYRBjCYxwOiv0I0VDTCZ4RHua+lhQNYjnJ8zMM60MTS+Q+vlgztW/GzEWSk2FqyeTkCrrJeJ/HozF0u+xK/GEQkZzRSYieDeDmPlhBNQni4RexE0IzKRIc8gkJcCnmmAimT7SJGMsMQFdt27Pzna1StoXVfuyat9flWu3aY95dIJOUQXZ6BrV0B1qohYi6Bm9oFf0Zrwbn8aX8b0YzRnpzhFagvHzC/8KrYA=</latexit> (Ef|Ei)C
|<latexit sha1_base64="Djo2uV7vzgXRINR18VDpO2fqyLM=">AAACO3icdZBNSwMxEIazflu/Wj2Jl2gRPJVdFfRYFMGjgrWCW0o2nW1Dk+ySzAplW/w1XvUX+EM8exOv3k1rD7biC4GXZ2bIzBulUlj0/TdvZnZufmFxabmwsrq2vlEsbd7aJDMcajyRibmLmAUpNNRQoIS71ABTkYR61D0f1usPYKxI9A32Umgo1tYiFpyhQ83idv+iGYeG6baEcDeUI0Md6zeLZb/ij0T/mmBsymSsq2bJK4SthGcKNHLJrL0P/BQbOTMouIRBIcwspIx3WRvundVMgW3koxsGdN+RFo0T455GOqK/J3KmrO2pyHUqhh07XRvC/2rYURO/55FhXcApFqmpFTE+beRCpxmC5j8bxpmkmNBhkrQlDHCUPWcYN8IdSXmHGcbR5e3SC6az+mtuDyvBUSW4Pi5Xz8Y5LpEdskcOSEBOSJVckitSI5w8kifyTF68V+/d+/A+f1pnvPHMFpmQ9/UNwsut2w==</latexit>EfihEf|
Figure 1: The basic setup for all processes in this work: An
agent with access to a systemS equipped with Hamiltonian
Hthat is assumed to be initially in thermal equilibrium with a
heat bath at inverse temperatureβsamples fromS(by
mea-suring in the energy basis), then implements a process that
maps the post-measurement state |EiihEi|toC(|EiihEi|),
whereCis a quantum channel. Finally, the agent repeats the
energy measurement onS with respect to the same
Hamil-tonianH.
which is the well-known Jarzynski equality (JE) for cyclic,
reversible processes [1]. Eq. (3)is strictly stronger than
(1), the latter being implied by(3)via Jensen’s inequality.
2.2
No macroscopic work
One of the reasons for the importance of the JE derives from the fact that it gives strong bounds on the possibil-ity of extracting work from a large system in a thermal
state [10,11,12]. To see this, letS be anN-particle
sys-tem and define the probability of extracting work w per
particle as
p(w) :=P(wN).
Plugging this into(3)yields that for any >0,
1 =heβWi=X
w
eβwNP(wN)≥eβNX
w≥
p(w),
which implies that events which extract significant posi-tive work per particle from a macroscopic system at
equi-librium are exponentially unlikely inN. For later use, we
formalize this property.
Definition 1 (No macroscopic work). Given a
se-quence ofN-particle systems initially at thermal
equi-librium with inverse temperature β and channels C
(implicitly depending on N), we say that the
pro-cesses represented byCfulfill theno macroscopic work
(NMW) condition if the probability of an event
ex-tracting work per particle larger or equal than is
ar-bitrarily small asN → ∞,
lim
N→∞p(w≥) := limN→∞ X
w>
p(w) = 0.
As is clear from the above, channels that satisfy the JE, such as unitary channels, also satisfy NMW and Av-SL. We now turn to investigate violations of JE and NMW for non-unitary channels.
3
Violations of NMW and JE
The first main result of this work is to introduce a
NMW and JE, but respects the Av-SL. To aid comparison, we first briefly discuss other generalizations of the stan-dard setting to non-unitary channels (see also Refs. [13, 14]).
3.1
Violating JE with non-unitary channels
It is easy to see from(2)that a more general class of
chan-nels that satisfy the JE areunitalchannels, that is, channels
that satisfyC[I] =I. Consequently, neither JE, nor in turn
NMW or Av-SL can be violated in settings which give rise to a unital channel. However, once this condition on uni-tality is relaxed, it becomes easy to violate JE on a formal level. For example, consider the fully-thermalizing
chan-nel that maps every input state to the thermal stateωβ(H),
in other wordsC(·) = ωβ(H). This channel always
vi-olates the JE whenever ωβ(H) 6= I/d. It is, however,
not clear how the energy-fluctuations can be interpreted
asworkin this example, since thermalizing processes
usu-ally occur due to contact with a heat bath, in which case one would naturally interpret the changes of energy on the system being due to heat. Thus, while it is trivial to for-mally violate JE, it is not obvious whether it is possible to do so in a physically meaningful and operationally
use-ful manner. Nevertheless, in AppendixA, we show that
the fully-thermalizing channel, in fact any channel with the thermal state as a fixed point, cannot violate the NMW condition for typical many-body systems, even if they may
violate(3). This means that, even if one interprets energy
fluctuations as work, one still could not use the thermaliz-ing channel to extract macroscopic amounts of work from a many-body system.
3.2
Violations of NMW and JE via
β
-catalytic
channels
The above findings raise the important question whether there exist channels for which the above procedure leads to a violation of NMW (and hence JE), while still respect-ing the Av-SL and allowrespect-ing for the interpretation of the
random variableWas work extracted fromS. Such
chan-nels, if they exist, promise to be of great interest because they could allow for a systematic exploitation of relatively likely events extracting work from heat baths. The first re-sult of this work is to answer this question affirmatively.
To this end, we define the notion of aβ-catalytic channel.
Definition 2 (β-catalytic channel). A completely
positive, trace-preserving mapCis aβ-catalytic
chan-nelon S, if there exists a quantum stateσC on a
sys-temC with Hamiltonian HC, together with a unitary
U such that [σC, HC] = 0and
C(·) = TrC(U(· ⊗ σC)U†),
s.t. TrS(U(ωβ(H)⊗σC)U†) =σC. (4)
Before stating our first main result, let us make some comments about this definition. First of all, we already assumed that the initial and final Hamiltonian coincides.
This means that while during the process,C may couple
system and catalyst for example by introducing
interac-tion termsHSC, nevertheless at the end of the process, the
channel must also turn off such interaction terms.
Sec-ondly, note that β-catalytic channels describe reversible
processes, in the sense that they do not change the entropy
of the joint-system SC and can be undone by acting on
this joint-system by a unitary process. We refer to the
sys-tem Cas being the “catalyst”, understanding that it may
be some by-stander system involving additional degrees of freedom. This terminology is motivated by the fact that, on average, i.e., if we do not condition on the outcomes of
the energy measurements, thenCis returned, at the end of
the procedure, to its original state. It can therefore be
re-used for further rounds of the protocol withnewcopies of
S. Note, however, that the invariance of the reduced state
onCunder the channel is required not for all initial states
of S, butonlyforωβ(H). As such,β-catalytic channels
depend onβandH through the second condition.
While Definition2 does not require the catalyst to be
uncorrelated with S at the end of the protocol, and in
this sense goes beyond the conventional notion of catal-ysis discussed in the resource-theoretic literature on
quan-tum thermodynamics [6, 7], the more general notion of
catalysis that we employ here is receiving increasing in-terest in quantum thermodynamics, where it was shown to single out the quantum relative entropy, free energy and
von Neumann entropy [15,8,16], to be useful in the
con-text of algorithmic cooling [16,17] and to show the
en-ergetic instability of passive states [18]. Finally, let us briefly comment on the interpretation of the random
vari-ableW as work in the setting ofβ-catalytic channels and
the role of the Hamiltonian of the catalyst. Since the
pro-cess onCandSis unitary, it is meaningful to denote the
total changes of energies of the two systems as work mea-sured by a two-point measurement scheme on each sys-tem. This gives rise to a joint-distribution of work on the
two systemsP(W(S), W(C)). The probability distribution
of workP(W)discussed above then simply corresponds
to the marginal distribution P(W(S))onS. Importantly,
this distribution is independent of the Hamiltonian onC
(see Sec. Gin the Appendix). In particular, we can
as-sume that the catalyst has trivial Hamiltonian HC = 0,
which in turn implies[σC, HC] = 0for anyσC. It is then
clear that no energy flows from the catalyst to the system, not even probabilistically. For the rest of the article, we
hence assume thatHC= 0.
Given these constraints, it may, at first glance, be un-clear how such a catalyst would offer any advantage to violating JE. For instance, one apparent way to make use
of the catalyst is to perform a controlled unitary onS,
con-ditioned onC: For someσC = Pipi|iihi|, one uses a
unitary in Eq.(4)of the form
USC:= X
i
Ui⊗ |iihi|C.
This special case ofβ-catalytic channels by construction
have the formCRU(·) =P
ipiUi(·)Ui†. But random
uni-tary channels are always unital, and therefore automati-cally satisfy JE.
In the following, we show that there exist non-unitalβ
-catalytic channels that allow for a meaningful violation of both NMW and JE, while at the same time they always respect the Av-SL. To see the latter, we note that these channels necessarily increase the von Neumann entropy of the input Gibbs state. This follows from the sub-additivity
of entropy and the fact thatCremains locally unchanged.
Now, sinceωβ(H)is the state with the least energy given
a fixed entropy [20,21], then we also have that
Tr(HC(ωβ(H))≥Tr(Hωβ(H))
which is just the Av-SL, concomitant with the findings of
Ref. [9]. We stress that despite this property,β-catalytic
channels are in generalnotunital. It remains to be shown
that β-catalytic channels that violate JE and NMW do
exist. We first show that JE can be violated already with small quantum systems, and then turn to the violation of NMW for macroscopic many-body systems with physically realistic Hamiltonians.
Microscopic violation of JE.As a toy-like example of
violating the JE withβ-catalytic channels, we consider a
system with three states – two degenerate (but
distinguish-able) ground states and an excited state with energyE. As
catalyst, we consider a system with two states and the uni-tary is a simple permutation between two pairs of energy
eigenvalues of the joint system (for details, see App.B). It
is straightforward to compute the probability distribution of work for such small systems, which in this case leads to
heβWi= Z+ 5 + 2(Z−2)(Z−1) Z(Z+ 1) ≥1,
whereZ = 2 + e−βE is the partition function of the
sys-tem and we used2 ≤Z ≤3. We hence findheβWi>1
whenever E > 0 (since then Z < 3) and we obtain a
moderate maximum violation in the limitE → ∞given
byheβWi= 7/6.
Macroscopic violation of NMW condition. We now
show that one can violate the NMW principle using cat-alysts.
Proposition 1 (Violation of no macroscopic work
with catalysts). Let (S(N))
N be a sequence of N
-particle locally interacting lattice systems with
Hamil-tonianH(N)that satisfy mild assumptions. Then, for
sufficiently largeN, there exist values of >0, such
that
p(w≥) (5)
can be brought arbitrarily close to 12 with β-catalytic
channels.
We provide a proof and full statement of the
assump-tions in Appendix D. Our assumptions are satisfied by
typical many-body Hamiltonians with energy windows in which the density of states grows exponentially [22].
While the formal proof of Proposition1is given in the
Appendix, the idea behind it is simple and we sketch it
here on a higher level. For a given N, let e(N) denote
the mean energy per particle of anN-particle system that
satisfies our assumptions. In the proof, we show that for
systems that satisfy the above assumptions and anyδ >0,
there exists anN and aβ-catalytic channelCsuch that
C(ωβ)≈δ
1
2 |E−ihE−|+ 1
2τ, (6)
where≈δdenotes equality of the states on LHS and RHS
up toδin trace distance, |E−iis some eigenvector ofH
with E− < e(N)N and τ is some other “fail”-state the
details of which are irrelevant. We can interpret Eq. (6)
as describing the approximation of a work extraction
pro-tocol that results in the state |E−iwith probability1/2.
Now, as the result of standard concentration bounds, for
largeN the mass of the thermal state ωβ will be highly
concentrated around energye(N)N. This implies that
ev-ery time the above work extraction protocol succeeds to prepare the ground state, for sufficiently high values of
N the extracted work per particle is arbitrarily close to
≡e(N)−E
−/N, leading to the statement of Prop.1.
We note that it is remarkable that catalytic channels, which are guaranteed to satisfy the Av-2nd law, allow for
the preparation of states like the one described in Eq.(6),
in which a pure low-energy state carries much of the weight, from a thermal state. Indeed, it has recently been
conjectured that with the help of catalystsanystate
tran-sition between full-rank states that increases the entropy
is possible [9], a statement known as thecatalytic entropy
conjecture. Prop.1, and in particular the ability to prepare
the state in Eq.(6), further supports this conjecture, which
has not been proven so far (even though strong evidence has been established).
Similar results as above also apply to the case in which
the initial state of the system is described by a
micro-canonical ensemblerather than the Gibbs state,
highlight-ing a similar contrast to fluctuation theorem results in the micro-canonical regime [23]. For detailed discussions and proves of corresponding statements in this regime, see
Ap-pendixC.
One may wonder whether the creation of correlations between system and catalyst is in fact necessary to violate the NMW principle. This is indeed true, when one simply forces the catalyst to remain uncorrelated in the definition
of β-catalytic channels. A proof of this statement along
with further discussion on this problem can be found in
AppendixI. Interestingly, such processes at the same time
allow for a violation of the Jarzynski equality. A particular example is given by the fully thermalizing channel, which can be realized using a catalyst that is simply a copy of the Gibbs state of the system and the unitary simply swapping the system and catalyst.
Required size of the catalyst. Proposition1not only
JE – the violation of JE is in fact exponential in the system
size. In particular,(5)implies that there exist values >0,
such that
heβWi ≥ 1
2e
βN 1
in the limit of largeN. It is natural to wonder how far the
JE can be violated and how big the catalyst has to be to re-alize a certain violation. This is clarified by the following result.
Proposition 2 (Bound on violation of JE). Let C be
any β-catalytic channel withdC= dim(HC). Then,
heβWi ≤min{dCkσk∞, dkωβ(H)k∞}
≤min{dC, d},
wherek · k∞ denotes the ∞-norm, which, for density
matrices, equals the largest absolute value of the in-put’s eigenvalues.
This proposition, the simple proof of which is given
in AppendixF, shows that in order to extract a growing
amount of work from a single run of a process, an external
agent will have to be able to prepare a stateσon a growing
auxiliary system and, more importantly, also have control over the increasingly large joint system. Hence, in prac-tice, the ability to violate JE will still be constrained by operational limitations. To illustrate the implications of
Prop.2, let us show how it immediately implies a bound
onP(W). As noticed when deriving the NMW principle,
for any≥0we have
heβWi ≥P(W ≥)eβ.
Hence, Prop.2implies
P(W ≥)≤dCkσk∞e−β.
In particular this means that to extract a macroscopic
amount of work, W ≥ wN, with finite probability, dC
has to grow exponentially withN (note thatkσk∞≤1).
4
Multi-partite work extraction
As emphasized before, even though the state of the cata-lyst remains unchanged in a catalytic process, in general it builds up correlations with the system. We now show that the correlations established between catalyst and sys-tem allow for processes in which many agents re-use the same catalyst to obtain highly inter-correlated work distri-butions.
Considernagents, each with identical systemsSi, i ∈
{1, . . . , n}that are initialized in the Gibbs stateω(β, H).
For a givenβ-catalytic channelCwith stateσon the
cat-alyst, consider the following protocol: Agent1 runs the
standard process from Fig.1using the catalyst and hence
implementingCbetween the two measurements. After the
procedure, she then passes C on to agent2 who repeats
this process, and so on, until the last agent has receivedC
and performed the process. From the catalytic nature ofC,
is is clear that, for each agent, the same marginal tion of work is obtained. However, the joint work distribu-tion for all agents will be correlated, due to individual
cor-relations between eachSiwithC. We now show that the
agents can use these correlations to systematically achieve certain global work distributions. Using the same notation
as before, letp(w1, . . . , wn)denote the global distribution
over the extracted work per particle, assuming that allSi
are copies of the same N-particle system. We have the
following, proven in AppendixE.
Proposition 3 (Multiple agents). Let each {Si}n
i=1 be a sequence of N-particle systems that satisfy the
conditions of Proposition 1. Then, for sufficiently
largeN, there exists an >0, such that
p(,−, ,−, . . .) =λ,
p(−, ,−, , . . .) = 1−λ, (7)
where λ can be brought arbitrarily close to 1/2 using
a sequence of β-catalytic channels onSi andC.
While(7)is clearly consistent with(1), this
proposi-tion shows that the agents can achieve joint work distri-butions that are strongly correlated and in which subsets of agents, in the above proposition one half of them, can violate JE arbitrarily, at the cost of the other half. Such distributions of work could, for example, be of interest in situations where the target is to maximize the probability that a subset of players extracts a positive amount work, at the ready cost of the others, for instance in order to surpass an activation energy. Importantly, the size of the catalyst
needed to realize the distribution(7)is fixed, i.e., it does
not scale with the number of agentsn.
Proposition3shows the existence of catalytic processes
that produce very interesting global work distributions. This naturally raises the question what other global dis-tributions can be obtained in a setting without making the size of the catalyst depend on the number of rounds. Our results, however, already imply that not every distribution compatible with the Second Law can be obtained in such
a way. For instance, Proposition2implies that the
distri-bution
p(, , , , . . .) =p(−,−,−,−, . . .)≈1/2
cannot be obtained viaβ-catalytic channels, since
other-wise there would exist a catalyst of fixed size that would
allow, for anyn, the total workW = nto be extracted
with probability approximately1/2, in violation of
Propo-sition2.
5
Summary and future work.
cyclic, reversible processes by introducing a catalyst—an additional system which, on average, remains unchanged after the protocol and can thus be re-used. This extension enables for distributions of work extraction that are not attainable without a catalyst. More precisely, one can by-pass the stringent conditions imposed by the JE, achieving positive work per particle with high probability, even for macroscopic systems. Furthermore, it allows for interest-ing, correlated work distributions when many agents use the same catalyst.
Our constructions illustrate in a striking way that the ab-sence of correlations, sometimes referred to as ‘stochastic independence’, can also be a powerful thermodynamic re-source [24]. This complements findings where the initial presence of correlations between a system and an ancilla are used to bypass the standard constraints imposed by
fluctuation theorems [25,26]. We discuss the connection
of our work to these findings in more detail in Appendix H. We believe that the further study of work distributions that can be obtained by collaborating agents by means of
β-catalytic channels will yield both foundational and
prac-tical insights.
We further believe that it is an interesting open prob-lem to study how the size of the catalyst has to scale if one wishes to maximize the probability to extract a certain amount of work. For example, in the context of a many-body system one might be content with extracting only an
amount of work of the order of √N if in exchange for
that one can either increase the probability for it to happen significantly or can reduce the size of the catalyst consid-erably (and hence the complexity of the unitary required to be implemented).
It would be interesting to understand the relation be-tween our results and a more generalized type of JE in the presence of information exchange [27], for example in a Maxwell demon scenario. In particular, in Ref. [28] it was also demonstrated that by using feedback control, one may also violate JE while respecting the Av-SL. More generally, our results also raise the question whether other phenomena –usually described as forbidden by the second law, or as occurring with vanishing probability– can be made to occur with high probability using catalysts. For example, is it possible to reverse the mixing process of two gases or induce heat flow from a cold to a hot system with finite probability in macroscopic systems? The tech-niques developed in this work provide a promising ansatz for the study of this and similar questions.
Acknowledgements. We thank Markus P. M¨uller and
Alvaro M. Alhambra for valuable discussions and
anony-mous referees for interesting comments. P. B.
ac-knowledges support from the John Templeton Founda-tion. H. W. acknowledges support from the Swiss Na-tional Science Foundation through SNSF project No. 200020 165843 and through the National Centre of
Com-petence in Research Quantum Science and Technology
(QSIT). N. H. Y. N. acknowledges support from the Alexander von Humboldt Foundation. R. G. has been sup-ported by the DFG (GA 2184/2-1). J. E. acknowledges
support by the DFG (FOR 2724), dedicated to quantum thermodynamics, and the FQXi.
References
[1] C. Jarzynski,Phys. Rev. Lett.78, 2690 (1997).
[2] G. E. Crooks,J. Stat. Phys.90, 1481 (1998).
[3] H. Tasaki, ArXiv e-prints (2000),arXiv:1303.6393.
[4] P. Erker, M. T. Mitchison, R. Silva, M. P. Woods,
N. Brunner, and M. Huber,Phys. Rev. X7, 031022
(2017).
[5] M. P. Woods, R. Silva, and J. Oppenheim,Ann Hen.
Poin.20, 125 (2019).
[6] F. G. S. L. Brand˜ao, M. Horodecki, N. H. Y. Ng,
J. Oppenheim, and S. Wehner, PNAS 112, 3275
(2015).
[7] N. H. Y. Ng, L. Manˇcinska, C. Cirstoiu, J. Eisert, and
S. Wehner,New J. Phys.17, 085004 (2015).
[8] M. P. M¨uller,Phys. Rev. X8(2018),
10.1103/phys-revx.8.041051.
[9] P. Boes, J. Eisert, R. Gallego, M. P. Mueller, and
H. Wilming,Phys. Rev. Lett.122, 210402 (2019).
[10] C. Jarzynski,Annu. Rev. Condens. Matter Phys. 2,
329 (2011).
[11] V. Cavina, A. Mari, and V. Giovannetti,Scientific
Rep.6, 29282 (2016).
[12] O. Maillet et al., Phys. Rev. Lett. 122, 150604
(2019).
[13] A. E. Rastegin,J. Stat. Mech.2013, P06016 (2013).
[14] A. E. Rastegin and K. ˙Zyczkowski,Phys. Rev. E89,
012127 (2014).
[15] H. Wilming, R. Gallego, and J. Eisert, Entropy19,
241 (2017).
[16] P. Boes, H. Wilming, R. Gallego, and J. Eisert,Phys.
Rev. X8, 041016 (2018).
[17] ´A. M. Alhambra, M. Lostaglio, and C. Perry,
Quan-tum3, 188 (2019).
[18] C. Sparaciari, D. Jennings, and J. Oppenheim,Nat.
Commun.8, 1895 (2017).
[19] K. M. R. Audenaert and S. Scheel,New J. Phys.10,
023011 (2008).
[20] A. Lenard,J. Stat. Phys.19, 575 (1978).
[21] W. Pusz and S. L. Woronowicz,Comm. Math. Phys.
58, 273 (1978).
[22] K. Huang,Statistical mechanics(Wiley, 1987).
[23] P. Talkner, P. H¨anggi, and M. Morillo,Phys. Rev. E
77, 051131 (2008).
[24] M. Lostaglio, M. P. M¨uller, and M. Pastena,Phys.
Rev. Lett.115, 150402 (2015).
[25] T. Sagawa and M. Ueda, Phys. Rev. Lett. 109,
180602 (2012).
[26] T. Sagawa and M. Ueda, New J. Phys.15, 125012
(2013).
[27] T. Sagawa and M. Ueda, Phys. Rev. Lett. 109,
180602 (2012).
[28] S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki, and
[29] M. Horodecki and J. Oppenheim,Nature Comm.4, 2059 (2013).
[30] C. Perry, P. ´Cwikli´nski, J. Anders, M. Horodecki,
and J. Oppenheim,Phys. Rev. X8, 041049 (2018).
[31] P. Faist, J. Oppenheim, and R. Renner,New J. Phys.
17, 043003 (2015).
[32] C. Gogolin, M. P. M¨uller, and J. Eisert,Phys. Rev.
Lett.106, 040401 (2011).
[33] A. Anshu,New J. Phys.18, 083011 (2016).
[34] M. A. Nielsen and I. L. Chuang,Quantum
Computa-tion and Quantum InformaComputa-tion (Cambridge Univer-sity Press, 2000).
[35] S. Goldstein, T. Hara, and H. Tasaki, ArXiv e-prints
(2013),arXiv:1303.6393.
[36] M. Kliesch, C. Gogolin, M. J. Kastoryano, A. Riera,
and J. Eisert,Phys. Rev. X4, 031019 (2014).
[37] J. Watrous, The theory of quantum information
(Cambridge University Press, 2018).
[38] M. Perarnau-Llobet, E. B¨aumer, K. V.
Hovhan-nisyan, M. Huber, and A. Acin,Phys. Rev. Lett.118,
070601 (2017).
[39] J. V. Koski, V. F. Maisi, T. Sagawa, and J. P. Pekola,
Phys. Rev. Lett.113, 030601 (2014).
A
NMW for Gibbs preserving maps
Thermalizing quantum maps, in particular those studied in the resource theoretic framework, are maps that model the evolution of a non-equilibrium quantum state as it ex-changes heat with its surrounding thermal bath. Several
variants of these maps exist [29,6,7,30,31], but a
com-mon feature is that they areGibbs preserving (GP), namely
that the Gibbs canonical state is a fixed point of such maps. Thermalizing maps are often viewed as “free operations” in a resource theoretic context, since they allow only for heat (instead of work) exchange with an environment in thermal equilibrium. In this section, we demonstrate two things: First, that even such thermodynamically “cheap” channels may violate the JE very strongly, due to non-unitality. Secondly, that they cannot be used to violate the NMW condition. A diagrammatic overview over the vari-ous properties of channels with respect to JE and NMW is
given in Fig.2.
We now turn to the first point. Given ad-dimensional
systemSwith HamiltonianH, the violation of JE can be
calculated for the thermalizing channel as
he−βWi=dX
j
e−βEj
ZH
hEj| C[I/d]|Eji,
=dX
j
e−βEj
ZH
hEj|ωβ(H)|Eji=
d deff
,
wheredeff := 1/Tr(ωβ(H)2)is known as the effective
di-mension [32] of the thermal state. One sees from the above
that JE is always violated forβ >0, sincedeff ≤d, with
equality only when ωβ(H) = I/d is maximally mixed.
For N non-interacting i.i.d. systems, both dandTr(ρ2)
scale exponentially withN, leading to an exponential
vio-lation inNfor JE.
Turning to the second point, one may wonder how this notion of thermodynamically free channels can be recon-ciled with the fact that JE is violated. However, note that in the standard JE setting, the work variable is traditionally defined in terms of a fluctuating (measured) energy dif-ference in the system, and does not inherently distinguish between work and heat contributions – unlike resource-theoretic settings where heat flow is allowed for free, but measurements incur a thermodynamic cost. Here, we con-sider an operationally more meaningful characterization (NMW as defined in Def. 1 of the main text), and show that NMW cannot be violated using channels that preserve the Gibbs state in generic many-body systems. The only assumptions that we make are that i) the system has
uni-formly bounded, local interactions on a D-dimensional
regular lattice and ii) a finite correlation length, i.e., the temperature is non-critical.
Lemma 3 (Non-violation of NMW for
Gibbs-pre-serving maps). No channelEthat preserves the Gibbs state can violate NMW for locally interacting many-body systems at a non-critical temperature.
Proof. We aim at showing that for any a > 0,
basic idea behind our proof is to make use of typ-icality. Let e(N) denote the energy density of the N-particle system and denote by Π(Nδ ) the projector onto energy eigenstates with energies in the interval TN,δ := [(e(N)−δ)N,(e(N)+δ)N]. Finally, denote
byp(·) the initial probability distribution of energy of the thermal stateτS(N), e.g., the probability that the initial energy measurement yieldsEi ∈ TN,δ is given
by
p(TN,δ) := Tr
τS(N)Π(Nδ ).
A theorem by Anshu [33] shows that under the given conditions most weight of the thermal stateτS(N) of theN-particle system is contained in a typical sub-space. More precisely, for a many-body system de-scribed by a D-dimensional lattice, there exist con-stantsC, K >0 such that for anyδ >0 we have
p(TN,δ)≥1−Ce−
(δ2N) 1 1+D
K . (8)
This is equivalent to saying that
p(TN,δc )≤Ce−(δ
2N)1+D1
K ,
where TN,δc = R\TN,δ. In particular, in the case of
D = 0, i.e., N non-interacting systems, we find the usual scaling obtained from Hoeffding’s inequality. In the following, for simplicity of notation, we writeσ1=
τS(N)and consider the normalized stateσ2obtained by
restrictingτS(N)to the subspace Π(N)δ as
σ2:=
Π(Nδ )τS(N) p(TN,δ)
.
Let us further write E(σ1(2)) =σ01(2), whereσ 0
1 =σ1
by assumption. Since the trace distance d(ρ1, ρ2) := 1
2Tr(|ρ1−ρ2|) fulfills the data processing inequality,
d(σ1, σ02) =d(σ 0
1, σ
0
2)≤d(σ1, σ2) =p(TN,δc ).
Using the operational meaning of trace distance d(ρ1, ρ2) = max
0≤M≤I|Tr(M(ρ1−ρ2))| [34], this means
that
|Tr(Π(Nδ )σ1)−Tr(Π
(N)
δ σ
0
2)| ≤p(T
c
N,δ) (9)
and, in turn,
Tr(Π(N)δ σ20)≥p(TN,δ)−p(TN,δc ) = 1−2p(T c
N,δ).(10)
To see this, note that (10) follows from (9) directly if Tr(Π(Nδ )σ0
2)≤Tr(Π
(N)
δ σ1), and as
Tr(Π(Nδ )σ02)>Tr(Π
(N)
δ σ1)≥Tr(Π
(N)
δ σ1)−p(TN,δc )
otherwise. This means that, conditioned on the fact that the initial state was within the typical energy
Jarzynski
Unital channels Average Second Law
- catalytic channels
No Macroscopic - Gibbs preserving
maps Equality
Work
Figure 2: A summary of different criteria (Av-SL, NMW and JE) mentioned in the main text, with examples of maps ac-cording to this characterization.
window (Ei∈TN,δ), the final energyEf is also within
this energy window except with probability 2p(TN,δc ), which is (sub-)exponentially small in N. We will use this later.
We are now ready to evaluate the probability of obtaining macroscopic work.
p(w≥a) =p(TN,δ)·p(w≥a|Ei∈TN,δ)
+p(TN,δc )·p(w≥a|Ei∈TN,δc )
≤p(w≥a|Ei∈TN,δ) +p(TN,δc ).
We can estimate the first term as
p(w≥a|Ei∈TN,δ)≤p(Ef ≤(e(N)+δ−a)N|Ei∈TN,δ).
We now choose δ=a/2 and get
p(w≥a|Ei∈TN,δ)≤p(Ef ≤(e(N)−a/2)N|Ei∈TN,δ)
≤Trhσ20I−Π(N)a/2i
≤2p(TN,a/2c ),
where we have used (10) in the last step. Altogether, we thus find
p(w≥a)≤3p(TN,a/2c ),
which decays to zero (sub-)exponentially by (8). This concludes the proof.
B
Microscopic toy example
In this section, we show that already for small systems and using catalysts, the JE can be violated. We do so by con-structing non-unital catalytic channels. Indeed, such maps can be realized “quasi-classically”, in the sense that in the construction it is sufficient to consider the energy spectra of the involved states and that all unitaries are simple per-mutations of those values. We consider a 3-level system
with energy levelsE1 = 0, E2 = 0, E3 = ∆in the
ther-mal state
w=
1
Z, 1 Z,
Z−2 Z
,
whereZ = 2 + exp(−β∆)is the partition function and
we express the state as a probability vector, such thatwi
denotes theith eigenvalue of the thermal state. For later,
we observe that2≤Z≤3.
We are going to construct a simple non-unital catalytic
channel that involves a 2-dimensional catalyst. Leteiand
fjdenote the basis states for the vector spacesVSandVC
describing the system and catalyst respectively. We define
the permutationπacting on the joint vector spaceVS⊗VC
as that permutation which exchanges the respective levels
e1⊗f1 ⇔ e2⊗f2ande2⊗f1 ⇔ e3⊗f2 and leaves
all other entries unchanged (see Fig.3). For the catalyst to
remain unchanged for this permutation and initial system state, it is easy to check that the catalyst has to be given by the vector
q=
Z−1
Z+ 1, 2 Z+ 1
.
Now, the catalytic channelCinduced by this catalyst and
permutation on the system has the general effect
C(p1, p2, p3) = (q1p2+q1p3, q1p1+q2p3, q2p1+q2p2),
so that, in particular, the maximally mixed input state is mapped to
C(I/3) = 2 3
Z
−1
Z+ 1, 1 2,
2 Z+ 1
,
which is different from the maximally mixed vector for
any∆>0.
What is more, we can also directly calculate the
work-distributionp(w), yielding
p(0) = 1
Z(Z+ 1)[Z+ 3 + 2(Z−2)(Z−1)], p(∆) = 2(Z−2)
Z(Z+ 1), p(−∆) = Z−1
Z(Z+ 1).
We now want to computeheβWi. To do so, it is useful to
note thate−β∆=Z−2and henceeβ∆= 1/(Z−2). We
find
heβWi= Z+ 5 + 2(Z−2)(Z−1) Z(Z+ 1) ≥1.
q2p3 q1p3 p3
q2p2 q1p2 p2
q2p1 q1p1 p1
q2 q1
→
q1p2 q1p3 q1p2+q1p3
q1p1 q2p3 q1p1+q2p3
q2p1 q2p2 q2p1+q2p2
q2 q1
Figure 3: We represent the joint state of system and catalyst
by means of a table. Left: At the beginning the joint system
starts out in a product state, so that the entry(i, j)is given
by the product of theith eigenvalue of the system andjth
eigenvalue of the catalyst. Right: After applying the
permu-tation highlighted in red, the marginal state of the system, given by the rows sums, has changed, while the marginal state of the catalyst (given by the column sums), has to re-main invariant. For a two-dimensional catalyst, specifying the permutation and initial system state fixes the catalyst state.
In fact, this quantity is larger than1wheneverZ <3,
cor-responding to ∆ > 0. Its maximum is given as 7/6for
Z = 3, which corresponds to∆→ ∞. Thus, the
Jarzyn-ski inequality is violated. At the same time the second law
is fulfilled as expected, sincep(−∆)≥p(∆).
C
Work extraction for initial
micro-canonical ensembles
In this appendix, we show that a statement similar to Proposition 1 of the main text holds in the slightly different setting of a micro-canonical initial state. This serves two purposes: i) in statistical mechanics, one often assumes that closed, macroscopic systems are described by micro-canonical ensembles due to the postulate of equal a priori probabilities of microstates corresponding to a macrostate. ii) The proof for the microcanonical initial state is con-ceptually simpler, but also provides the blueprint for the slightly more involved proof in the case of a canonical
state, which is provided in Sec.D.
In the following, we denote by I ⊂ Ran energy
win-dow, byg(I)the number of energy eigenstates in this
win-dow,
g(I) = X
Ei∈I
1,
and the corresponding micro-canonical state by
ΩS(I) =
1 g(I)
X
Ei∈I
|EiihEi|.
A micro-canonical energy window around energy density
eis any energy windowI(e)of the form[e−O(√N), e],
whereNis the number of particles.
The only difference to the standard setting described in the main text (as depicted in Fig. 1) is that the initial state
differs from the thermal stateωβ(H). Instead, it is given
by the micro-canonical ensemble. In other words, given a
Cof the form
C(·) = TrC(U(· ⊗σC)U†)
s.t.TrS(U(ΩS(I)⊗σC)U†) =σC.
We carry over notation from the main text, so thatp(w≥
)denotes the probability of measuring the system’s
en-ergy per particle decrease by at least an amount, and so
on. Furthermore, we take the catalyst Hamiltonian in our
construction to beHC=I.
We will now first show that the NMW principle also holds for micro-canonical states of generic many-body systems. After that we will show that it can be circum-vented using catalysts. To show the validity of the NMW principle we will use the same reasoning as presented in Ref. [35], where the NMW principle has been studied be-fore. Thus, the following proof is essentially a reproduc-tion for the convenience of the reader. We consider a
se-quence of many-body HamiltoniansHS(N)onN particles
with the generic property of having an exponential density of states:
g((−∞, E]) := X
Ei≤E
1 = eN µ(E/N)−o(N), (11)
whereµis a strictly monotonic and differentiable function
independent ofNando(N)denotes terms small compared
toN,limN→∞o(N)/N= 0.
Proposition 4 (NMW for micro-canonical states).
Consider a sequence of N-particle Hamiltonians
ful-filling(11)and a sequence of micro-canonical
energy-windows I(N)= [eN, eN+δ√N] around energy
den-sitye(withδ >0 fixed). Then for any unital channel
acting on theN-particle system, the probability of
ex-tracting workw per particle is bounded as
p(w > )≤Ce−µ0(e)N+o(N),
whereC >0 is a constant andµ0 denotes the
deriva-tive ofµ.
Proof. LetI≤ := (−∞,(e−)N +δ
√
N], denote by PS(I≤) the projector onto energy-eigenstates with
en-ergies below (e−)N+δ√Nand letU denote a unital channel. In the following, we write I instead ofI(N)
to simplify notation. Then
p(w > )≤Tr (PS(I≤)U[ΩS(I)])
= X
Ei∈I
1
g(I)Tr (PS(I≤)U[|EiihEi|])
≤ 1
g(I)Tr (PS(I≤)U[I]) = g(I≤)
g(I) . Writing ˜e:=e+δN−1/2, we have
g(I) = eN µ(˜e)−o(N)−eN µ(e)−o(N)
= eN µ(˜e)−o(N)1−e−N(µ(˜e)−µ(e))+o(N)
≈eN µ(˜e)−o(N),
where in the last estimation we use that µis strictly monotonic. In particular, we can estimate the expo-nential in the parenthesis as
e−N(µ(˜e)−µ(e))−o(N)=Oe−δµ0(e)N1/2,
where µ0 denotes the derivative ofµ. Using g(I≤) =
eN(µ(˜e−)−o(N) we then find
p(w > )≤ e
−N(µ(˜e)−µ(˜e−))+o(N)
1−O(e−δµ0(e)√N
)
≤Ce−µ0(e)N.
We have here used thatµis differentiable to prove this
result. Similar results would follow for weaker notions
of regularity of µ, such as Lipschitz-continuity. Having
proven the NMW principle for generic many-body sys-tems, let us now show how to circumvent it using catalysts.
Proposition 5 (Overcoming NMW using catalysts).
Consider a Hamiltonian HS and a microcanonical
stateΩS(I), with I a micro-canonical energy window
around energy density e. Suppose there exists an
en-ergy window I+ with g(I+) = g(I)2. Then, for any
0≤e−< e, there exists a catalytic channel such that
p(w≥e−e−) =
1 2.
Before giving the proof of the proposition, let us em-phasize again that the required conditions on the Hamilto-nian are very weak. In particular, the conditions are (proximately) fulfilled if the density of states is well ap-proximated by an exponential in the range of energies that we are working in, a condition that is typically fulfilled in many-body systems and, as we have seen above, leads to an NMW principle if we do not allow for catalysts.
Proof. A sketch of the proof is given in Fig. 4. The
proof is constructive in the sense that we provide an explicit catalyst and unitary. We first introduce some useful notation. Define g :=g(I),g+ :=g(I+) = g2
and let PS(I) and PS(I+) be the projectors onto the
corresponding energy subspaces. Let |E−i be any
eigenstate of the Hamiltonian such that 0≤E−/N=
e− ≤ e. Following this notation, the initial state of
the system is
ΩS(I) =
1 gPS(I).
The aim is to bring the system to a state that is an equal mixture of |E−ihE−| and Ω(I+). To
do this, we employ a catalyst of dimension dC =
g + 1. Let { |iiC}dC
i=1 be an arbitrary orthonormal
basis on the Hilbert-space of the catalyst and let PC =
Pg
i=1|iihi|. The initial state on the catalyst
is given by
σ= 1 2gPC+
1
